On main eigenvalues of certain graphs

TL;DR

This paper investigates the properties of main eigenvalues in certain graphs, characterizes harmonic graphs in terms of these eigenvalues, and identifies conditions for specific eigenvalues in graph complements, focusing on paths, double stars, and bipartite graphs.

Contribution

It provides new characterizations of main eigenvalues, especially for harmonic graphs, and establishes conditions for eigenvalues of graph complements, including a unique property of complete bipartite graphs.

Findings

Graphs with exactly two main eigenvalues are characterized.

Harmonic graphs are characterized by their main eigenvalues.

Paths and double stars with non-main smallest eigenvalues are identified.

Abstract

An eigenvalue of the adjacency matrix of a graph is said to be \emph{main} if the all-1 vector is not orthogonal to the associated eigenspace. In this work, we approach the main eigenvalues of some graphs. The graphs with exactly two main eigenvalues are considered and a relation between those main eigenvalues is presented. The particular case of harmonic graphs is analyzed and they are characterized in terms of their main eigenvalues without any restriction on its combinatorial structure. We give a necessary and sufficient condition for a graph $G$ to have $-1-\lambda_{\min}$ as an eigenvalue of its complement, where $\lambda_{\min}$ denotes the least eigenvalue of $G$. Also, we prove that among connected bipartite graphs, $K_{r,r}$ is the unique graph for which the index of the complement is equal to $-1-\lambda_{\min}$. Finally, we characterize all paths and all double stars (trees with diameter three) for which the smallest eigenvalue is non-main. Main eigenvalues of paths and double stars are identified.